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Learning Commons

Quantitative & Symbolic Reasoning

Central Washington University Quantitative and Symbolic Reasoning Skills and Sample Items

Quantitative and symbolic information may be displayed visually in graphs or charts or in numerical form using whole numbers, fractions, decimals, percentages, or time units (hours and minutes). These quantities appeared in both prose and document form. This document will illustrate the standards with sample items.

A full document with more sample problems may be found here.

Goal 1: Have a very firm grasp of "Number Sense" and Algebra.

This goal includes skills and concepts that we expect incoming CWU students to have already mastered. Our remediation program is for the students have not mastered these skills.

Be competent in basic arithmetic.

Example: Twenty-five of the 65 students ate in the cafe yesterday. What percentage of the students ate in the cafe yesterday?

Interpret and use simple numerical information.

Example: Chang put his $5000 spring break vacation on his credit card over 7 years ago. He has been paying the credit card company $120 every month for the past 84 months in order to pay off his vacation. Calculate the total cost of Chang's vacation.

Perform basic estimation tasks.

Example: When the wind is blowing on a cold day, the temperature seems to feel colder than when the air is still. Meteorologists refer to this cooling power of moving air as "wind chill," and use it to describe the relationship between the actual temperature and the temperature at which the air appears to be when the wind is blowing.

Estimate the "wind chill" temperature when the thermometer says it is 35° and the wind is blowing 22 mph.

What wind speed do you estimate is needed to produce a "wind chill" temperature of 0° F.

Wind Chill Effects at 35 Degrees
Wind Speed (mph)Equiv. Air Temp. (degrees)


Perform basic algebraic and symbolic manipulations.

Solve for x:

One half, times the quantity X plus 40, plus 1, equals 70


Goal 2: Be able to understand, analyze, and interpret quantitative information from a variety of sources (graphs, calculations, statistics and symbols).

Our students must be able to understand and analyze information from many sources to make sense of many health, social, economic, technological and environmental problems. This reasoning involves representing, analyzing, and solving problems in numeric, graphical, and symbolic contexts.

Read and generate a variety of graphs.

Example: What percentages of the nations health care expenditures did the government pay in Medicare and other public assistance programs in the year 2000?

Example: Explain what this percentage means to the health care system and use a pie chart to support your explanation.

Example: Using the given information, to predict if our government's percentage of health care expenditures will increase, decrease, or remain the same. Support your conclusions with an appropriate graph.

National Health Care Expenditures (in billions of $)
National Health Expenditures6969901,2161,299
Public Assistance Medical Payments79150192208


Interpret basic statistical summaries.

The mean salary of the nine employees of the World Wide Education Co. is $30,889.

  • The CEO makes $100,000 per year
  • Two managers make $50,000 per year
  • Four factory workers make $15,000 each
  • Two trainees make $9,000 per year

Example: Use the mean and actual salaries of the nine employees to explain why the mean salary is much better than most of the employees experience?

Explain the limitations of statistics.

"Good morning. You are listening to CWUX radio. It's election day! This morning, the Journal-Reporter reported that the latest polls indicate Republican Bennett leading Democrat incumbent Cummins by a margin of 7%, with a margin of error of +/- 5%. If these numbers hold, Bennett will be the city's new mayor." Example: Explain how these figures were generated, what they really mean. Explain the limitations of deductive reasoning. I saw the same women leave that office building every day at 5 PM. She must work in the building.

Example: Explain whether this is inductive or deductive logic and if a valid argument can be made for this conclusion.

Goal 3: Be able to represent and understand the representations of real-world problems using such abstractions as symbols and graphs.

Our students must be able to successfully use this quantitative and symbolic reasoning to represent and solve real-world problems. The use of abstract representations such as symbols and graphs make it possible to analyze the complex problems we find in the real-world.

Identify problems in context.

Example: Explain how you could use the census table to estimate the future US population.

Express contextual problems in abstract and symbolic form.

Example: Create an algebraic model (equation) to estimate the future US population.

Interpret functions, functional notation, graphs, and quantitative data in a specific context.

Example: Use the 1990 and 2000 census to verify your population model (equation).

Distinguish between and interpret a variety of real world models.

Example: What type of model (equation) best-fits the given data?

  1. linear
  2. quadratic
  3. exponential
  4. logistic




Identify and use deductive reasoning in a logical argument.

A politician argued that statistics show that a criminal offense occurs every 2 seconds, violent crimes occur every 16 seconds, and robberies occur every 48 seconds in the US. Clearly this means we need to increase the conviction rate of offenders and strengthen police forces.

Example: Write an evaluation of this argument by identifying the conclusion and identify the stated, unstated, and assumed premises.